Question

If $f(x)=\left|\begin{array}{ccc}\sin x& \cos x& \tan x\\ x^3& x^2& x\\ 2x& 1& x\end{array}\right|$ then $\underset{x\to 0}{\mathrm{Lim}}\frac{f(x)}{x^2}$ is equal to

• A. 0
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (D)

Given limit $=\underset{x\to 0}{\mathrm{Lim}}\left|\begin{array}{ccc}\frac{\sin x}{x}& \cos x& \frac{\tan x}{x}\\ \frac{x^3}{x}& x^2& \frac{x}{x}\\ \frac{2x}{x}& 1& \frac{x}{x}\end{array}\right|=\left|\begin{array}{ccc}1& 1& 1\\ 0& 0& 1\\ 2& 1& 1\end{array}\right|$
$=1(0-1)+2(1-0)=1$.